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How coordinates are defined in semimetric spaces? Imagine this example:

We have points A, B and C.

We know the distances between this points:

d(A,B) = 10,381
d(A,C) = 3,896
d(B,C) = 3,896

This is a semimetric space because

d(A,B) > d(A,C) + d(B,C)

What does that mean then? How do we express the coordinates of this space?

We could find some coordinates that make the previous distances true for A and B:

A = (0, 0)
B = (0, 10.896)

But how would we express the position of C?

There's no way to represent the coordinates of a set of points in a semimetric space?

Thanks

Pau Rosello
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  • "Coordinates" are typically used to represent points in vector spaces, with respect to some basis. Not every (semi)metric space is a vector space, and I don't know how you talk about coordinates outside of the setting of a vector space. – Xander Henderson Nov 15 '21 at 02:07
  • So some semimetric spaces can be vector spaces? – Pau Rosello Nov 15 '21 at 02:08
  • I have not thought deeply about the question of whether or not a semimetric space can be a vector space (if, for no other reason, I don't know what your definition of a semimetric space is, or if you consider a metric space to be a special case of a semimetric space). My instinct is that you are going to have trouble defining a semimetric on a vector space which is both compatible with the vector space structure and not a metric. – Xander Henderson Nov 15 '21 at 02:13
  • I consider a semimetric space a space that does not satisfy the triangle inequality. How do we express positions in this kind of spaces? – Pau Rosello Nov 15 '21 at 02:37

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